At the heart of every medical imaging technology is a sophisticated mathematical model of the measurement process and an algorithm to reconstruct an image from the measured data. This book provides a firm foundation in the mathematical tools used to model the measurements and derive the reconstruction algorithms used in most of these modalities. The text uses X-ray computed tomography (X-ray CT) as a 'pedagogical machine' to illustrate important ideas and its extensive discussion of background material makes the more advanced mathematical topics accessible to people with a less formal mathematical education. This new edition contains a chapter on magnetic resonance imaging (MRI), a revised section on the relationship between the continuum and discrete Fourier transforms, an improved description of the gridding method, and new sections on both Grangreat's formula and noise analysis in MR-imaging. Mathematical concepts are illuminated with over 200 illustrations and numerous exercises.

Medical imaging is a major part of twenty-first century health care. This introduction explores the mathematical aspects of imaging in medicine to explain approximation methods in addition to computer implementation of inversion algorithms.

Biomedical imaging is a fascinating research area to applied mathematicians. Challenging imaging problems arise and they often trigger the investigation of fundamental problems in various branches of mathematics. This is the first book to highlight the most recent mathematical developments in emerging biomedical imaging techniques. The main focus is on emerging multi-physics and multi-scales imaging approaches. For such promising techniques, it provides the basic mathematical concepts and tools for image reconstruction. Further improvements in these exciting imaging techniques require continued research in the mathematical sciences, a field that has contributed greatly to biomedical imaging and will continue to do so. The volume is suitable for a graduate-level course in applied mathematics and helps prepare the reader for a deeper understanding of research areas in biomedical imaging.

Covering the basics of X-rays, CT, PET, nuclear medicine, ultrasound, and MRI, this textbook provides senior undergraduate and beginning graduate students with a broad introduction to medical imaging. Over 130 end-of-chapter exercises are included, in addition to solved example problems, which enable students to master the theory as well as providing them with the tools needed to solve more difficult problems. The basic theory, instrumentation and state-of-the-art techniques and applications are covered, bringing students immediately up-to-date with recent developments, such as combined computed tomography/positron emission tomography, multi-slice CT, four-dimensional ultrasound, and parallel imaging MR technology. Clinical examples provide practical applications of physics and engineering knowledge to medicine. Finally, helpful references to specialised texts, recent review articles, and relevant scientific journals are provided at the end of each chapter, making this an ideal textbook for a one-semester course in medical imaging.

Fundamentals of Medical Imaging, second edition, is an invaluable technical introduction to each imaging modality, explaining the mathematical and physical principles and giving a clear understanding of how images are obtained and interpreted. Individual chapters cover each imaging modality – radiography, CT, MRI, nuclear medicine and ultrasound – reviewing the physics of the signal and its interaction with tissue, the image formation or reconstruction process, a discussion of image quality and equipment, clinical applications and biological effects and safety issues. Subsequent chapters review image analysis and visualization for diagnosis, treatment and surgery. New to this edition: • Appendix of questions and answers • New chapter on 3D image visualization • Advanced mathematical formulae in separate text boxes • Ancillary website containing 3D animations: www.cambridge.org/suetens • Full colour illustrations throughout Engineers, clinicians, mathematicians and physicists will find this an invaluable aid in understanding the physical principles of imaging and their clinical applications.

This book surveys the main mathematical ideas and techniques behind some well-established imaging modalities such as X-ray CT and emission tomography, as well as a variety of newly developing coupled-physics or hybrid techniques, including thermoacoustic tomography. The Radon Transform and Medical Imaging emphasizes mathematical techniques and ideas arising across the spectrum of medical imaging modalities and explains important concepts concerning inversion, stability, incomplete data effects, the role of interior information, and other issues critical to all medical imaging methods. For nonexperts, the author provides appendices that cover background information on notation, Fourier analysis, geometric rays, and linear operators. The vast bibliography, with over 825 entries, directs readers to a wide array of additional information sources on medical imaging for further study.

At the heart of every medical imaging technology is a sophisticated mathematical model of the measurement process and an algorithm to reconstruct an image from the measured data. This book provides a firm foundation in the mathematical tools used to model the measurements and derive the reconstruction algorithms used in most imaging modalities in current use. In the process, it also covers many important analytic concepts and techniques used in Fourier analysis, integral equations, sampling theory, and noise analysis.This text uses X-ray computed tomography as a "pedagogical machine" to illustrate important ideas and incorporates extensive discussions of background material making the more advanced mathematical topics accessible to readers with a less formal mathematical education. The mathematical concepts are illuminated with over 200 illustrations and numerous exercises.New to the second edition are a chapter on magnetic resonance imaging (MRI), a revised section on the relationship between the continuum and discrete Fourier transforms, a new section on Grangreat's formula, an improved description of the gridding method, and a new section on noise analysis in MRI. Audience The book is appropriate for one- or two-semester courses at the advanced undergraduate or beginning graduate level on the mathematical foundations of modern medical imaging technologies. The text assumes an understanding of calculus, linear algebra, and basic mathematical analysis. Contents Preface to the Second Edition; Preface; How to Use This Book; Notational Conventions; Chapter 1: Measurements and Modeling; Chapter 2: Linear Models and Linear Equations; Chapter 3: A Basic Model for Tomography; Chapter 4: Introduction to the Fourier Transform; Chapter 5: Convolution; Chapter 6: The Radon Transform; Chapter 7: Introduction to Fourier Series; Chapter 8: Sampling; Chapter 9: Filters; Chapter 10: Implementing Shift Invariant Filters; Chapter 11: Reconstruction in X-Ray Tomography; Chapter 12: Imaging Artifacts in X-Ray Tomography; Chapter 13: Algebraic Reconstruction Techniques; Chapter 14: Magnetic Resonance Imaging; Chapter 15: Probability and Random Variables; Chapter 16: Applications of Probability; Chapter 17: Random Processes; Appendix A: Background Material; Appendix B: Basic Analysis; Index.